Gaming Guru

In the Casino, Seeing Shouldn't Always Be Believing

Theories abound as to what makes cards, dice, and pictures on slot machine
reels and screens exhibit the trends and patterns many bettors know they
see and believe occur. Gamblers reinforce their beliefs by watching for
supporting instances, taking these events as evidence and ignoring
everything else.

Among my favorite examples is the notion that "10s follow 10s" in
blackjack. That is, if a 10-valued card appears, the next card is more
likely than usual to be another 10. Carried further, this theory implies
that runs or clumps of successive 10s occur more often than predicted
strictly by the laws of probability.

Bettors who believe 10s follow 10s are sure they're right because they
see it with their own eyes. They've heard that chances of five, six, or
more 10s in a row are minuscule. But they've played plenty of blackjack
and noticed numerous runs like these. And dealers tell them they observe
it "all the time." Some solid citizens embrace the hypothesis without
trying to explain it, just as they accept microwave cooking without
wondering what gets the potato hot. Others think 10s follow 10s because
of imperfect shuffles as well as the order in which cards are picked up
and placed in the discard rack during and after a round.

It's a
fact that the chance of drawing a long series of successive 10s is quite
small. It's also a fact that players -- and to a greater extent, dealers
-- will see them fairly often. These two facts aren't contradictory.
Taken together, they indicate why occurrences of clumps won't help beat
the odds.

To picture what I mean, assume cards are drawn from an infinitely large
shoe. The laws of probability state that the chances a random draw will
yield four 10s in a row are roughly 9 out of 1,000. The chances of five
and six 10s in a row are under 3 out of 1,000 and 1 out of 1,000,
respectively. Pretty low.

So, assume that in the course of an hour's play, you observe a run of
six or more 10s in a row. And, further, that you notice runs at least
this long almost every time you play. Wouldn't you be tempted to
conclude that since the chances of such a thing are so small, your
observing it regularly proves something about 10s following 10s and the
clumping of cards in the shoe?

You'd be wrong. During an hour, you're apt to see 1,250 cards. And a run
of six 10s could occur anywhere in that set of cards -- from positions
1-2-3-4-5-6, 2-3-4-5-6-7, 3-4-5-6-7-8, through
1243-1244-1245-1246-1247-1248, 1244-1245-1246-1247-1248-1249,
1245-1246-1247-1248-1249-1250. The chance of seeing that 1 out of 1,000
event is increased by the possibility that it can occur at any of 1,245
positions. Added to this, you'd likely believe the theory verified if
you observed sequences such as five rather than six 10s in a row, or six
out of seven successive cards being 10s. All told, the chance of seeing
something you think confirms the theory can get fairly high, simply by
virtue of the many ways for the individually rare events to occur and
independently of anything special about 10s following 10s or clumping.

Here are some specific figures for perspective. The chance of at least
one clump of 10s longer than five cards in a series of 1,250 completely
random draws is almost 40 percent. That is, you can expect to see one or
more such runs in four out of every 10 hours you play. The chance of
seeing at least one run longer than six cards in an hour of play exceeds
14 percent. And the chance of one or more runs over seven cards in
length is over 4 percent.

How can this information help you? It arm you with knowledge. By
rejecting spurious theories, you're better able to resist the urge to do
something silly after seeing a few 10s in a row, falsely anticipating an
unusually high chance of drawing 10 again. Long runs of 10s are random
phenomena, which are likely during a game by chance alone. No clumping
mechanism induces them to happen. You therefore can't predict when
they'll occur or how long they'll last. As the poet, Sumner A Ingmark,
whose random runs of rhymes never forecast what's to follow, reminded
readers:

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